Design of a MEMS-based 52 MHz oscillator

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Citation for published version (APA):

Bontemps, J. J. M. (2009). Design of a MEMS-based 52 MHz oscillator. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR644305

DOI:

10.6100/IR644305

Document status and date: Published: 01/01/2009 Document Version:

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Design of a MEMS-based 52 MHz

oscillator

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Print: Universiteitsdrukkerij Technische Universiteit Eindhoven, The Netherlands. Cover design: Klasien Visser.

Cover: The cover shows a MEMS free-free beam resonator processed in 1.5 μm SOI. The beam has a length of 8 μm and width of 1 μm, which results in a resonance frequency of 405 MHz.

A catalogue record is available from the Eindhoven University of Technology Library.

Bontemps, Joep Jacques Marie

Design of a MEMS-based 52 MHz oscillator / by Joep Jacques Marie Bontemps. – Eindhoven : Technische Universiteit Eindhoven, 2009. – Proefschrift.

ISBN: 978-90-386-1956-9 NUR: 926

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Design of a MEMS-based 52 MHz

oscillator

PROEFONTWERP

ter verkrijging van de graad van doctor aan de

Technische

Universiteit

Eindhoven, op gezag van de

rector magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie

aangewezen

door het College voor

Promoties in het openbaar te verdedigen

op maandag 21 september 2009 om 16.00 uur

door

Joep Jacques Marie Bontemps

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De documentatie van het proefontwerp is goedgekeurd door de promotoren:

prof.dr. H.C.W. Beijerinck

en

prof.dr. P.J. French

Copromotor:

dr. J.I.M. Botman

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CHAPTER 1 INTRODUCTION

1.1 MICRO ELECTRO-MECHANICAL SYSTEMS (MEMS)

In the late ‘50’s Jack Kilby and Robert Noyce were both working on the same question: “how to make more of less?” In 1952, Geoffrey Dummer had been the first to conceptualise the integrated circuit. Continued efforts by various people eventually led to the invention of the first working IC in 1958 by Jack Kilby, who worked for Texas Instruments at the time. In 1959, Robert Noyce (Fairchild Semiconductors) independently invented his own working IC combining work of Hoerni and Lehovec. Where Kilby used germanium, Noyce used silicon. Although he was half a year later, Noyce’s chip had solved many practical problems that the microchip developed by Kilby had not [1]. The two men could not, however, have foreseen that this marked the start of the semiconductor industry with a global market worth about $250 billion in 2008 [2]. Over the past fifty years microelectronics has had a large impact in health, communication, entertainment, and most other areas of modern life.

The technological progress in the early years of the semiconductor industry has been described by Moore in 1965. He found that the complexity of integrated circuits, defined as

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the number of components per chip, had approximately doubled every year since their introduction [3]. In 1975 he predicted that the rate of increase would slow down in the next years to approximately a doubling every two years [4]. This exponential growth in complexity is called Moore’s Law and the trend is observed until today.

The question how to make more of less remains relevant today. It cannot only be achieved through miniaturization (Moore’s Law), but also by function integration. Semiconductor industry has always looked for ways to increase functionality of the IC by adding non-integrated components. Micro-Electro-Mechanical Systems (MEMS), also commonly referred to as micromachining in Japan or microsystems in Europe, is an enabling technology that can significantly increase the functionality of an IC.

MEMS is the integration of mechanical elements, sensors, actuators, and electronics on a common silicon substrate through micro-fabrication technology. While the electronics are fabricated using IC-process sequences, the micromechanical components are fabricated using compatible "micromachining" processes, e.g. deposition, lithography and etching. Silicon parts are selectively etched away or new structural layers are added to form the mechanical and electromechanical devices.

FIGURE 1.1 – In (a), a MEMS clamped-clamped beam resonator is depicted [5]. The beam measures 50 μm across. Figure (b) shows a MEMS ratchet mechanism [6], a mechanical device that controls the rotational motion.

MEMS interface the computing power of the electronic world with the sensor and actuators of the non-electronic world. Microelectronic ICs can be thought of as the "brains" of a system and MEMS augments this decision-making capability with "eyes" and "arms", to

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allow microsystems to sense and control the environment. At first, the MEMS field mainly covered electromechanical transducers. The breakthrough of MEMS systems has been triggered by the development of accelerometers for application in airbags in the automobile industry. Today, MEMS applications cover a wider range. A list of examples is given in Table 1.1.

TABLE 1.1 – Examples of applications in the wide field of MEMS [7].

Category Examples

Inertia sensors Accelerometer (airbag), gyroscope

Pressure sensors Tire pressure, blood pressure, microphone

Micro fluidics / bioMEMS Inkjet printer nozzle, micro-bio-analysis systems, DNA chips

Optical MEMS (MOEMS) Micro-mirror array for projection, optical fibre switch, _{adaptive optics} RF MEMS Switches, Antenna-filter, high-Q inductor

The topic of this thesis is silicon MEMS resonators and oscillators. MEMS resonators are mechanical structures that vibrate on-chip. With their high-Q eigenmode they serve as frequency filter in filters or oscillator applications.

1.2 THE TIMING MARKET

1.2.1 Market size and development

In 2008, timing devices constituted a $4.1 billion market [8]. The predicted annual growth rate is 6-7%. The combination of large size and large growth rate makes it a very attractive market. Currently, 92% of the revenue is accumulated in Asia. The market can be divided into:

- CMOS oscillators - MEMS

- Crystal oscillators

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The market share (revenues) of each product type is depicted in Fig. 1.2. The 57% market share of the crystal oscillators is larger than the rest of the market combined. The average selling price (ASP) of a timing device is $0.15. Important to note is that within this market the ASP of timing IC’s is $1.25. Hence, the IC adds considerable value.

FIGURE 1.2 – The worldwide timing devices revenue in 2008 accumulates to $4.1 billion. The figure [8] shows the market share per product type: 57% is held by crystal oscillators, while timing IC’s account for 43%. CMOS oscillators are hardly sold as separate devices.

Size Price Time Performance: - Package - Power - Precision - Phase noise

FIGURE 1.3 – The price of a timing device is determined by its size and performance. Prices erode over time. An increase in performance, i.e. package, power, precision, and phase noise as indicated by the 4 P’s, increases the price of a timing device of specific size.

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Currently, there are over a hundred significant producers of crystal oscillators. Quartz crystals (resonators) and crystal oscillators are found in most modern electronic equipment. Frequencies for the resonators and oscillators range from kHz to hundreds of MHz.

Crystal oscillators are used as clock, tuning device, timing reference or frequency converter. Typical applications include:

Computer (41%)

- PC motherboards, memory modules Consumer (28%)

- Set-top boxes, DVD players and recorders, Video game consoles, digital camera’s, watches, AM/FM radio

Communication (26%)

- Telecommunication, wireless and broadband, wired communication Automotive and Industrial (2% and 3% respectively)

For a crystal oscillator, there is a strong correlation between size and price. Higher prices are paid for smaller oscillators. Furthermore, a higher price can be demanded for an oscillator with superior performance. The performance is described by the four P’s of oscillators: Package, Power, Precision (accuracy), and Phase noise. Package stands for the compatibility with the system and not the package size. As an example, a package with solder balls instead of pins can often result in a significant prize premium. For the customer, the package with solder balls reduces assembly time and consequently cost.

1.2.2 Quartz or silicon?

Due to their high Q-factor, mechanical resonators are preferred over electrical resonators in high-precision filters for time-keeping and frequency-reference applications. For decades, crystal oscillators have set the standard in the timing market. Next to the high Q-factor, quartz crystals offer two main advantages. First, quartz is piezoelectric. This effective transduction principle leads to resonators with low signal losses. Second, quartz cut along specific directions shows almost zero temperature drift. This results in highly accurate resonators over a typical temperature range of 100 ºC.

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FIGURE 1.4 – The main board of a Sony Walkman NW-HD1 showing three discrete timing devices and a mechanical acceleration sensor (photo: Portelligent).

2008 2009 2010 2011 2012 2013 0.00 0.20 0.40 0.60 AS P ($ ) 0 100 200 _MU SD MU

FIGURE 1.5 – Worldwide MEMS resonator market forecast. The figure shows predictions of the average selling price (ASP) and the total sales volume in million units (MU) and million US dollars (MUSD) over the next five years.

HDK #HAAM-301BS

Piezoresistive Acceleration Sensor

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Crystal oscillators offer a combination of excellent phase noise performance, low power, and high accuracy unmatched by other timing devices. However, quartz crystals are also bulky and are currently not integrated with IC technology. MEMS resonators have been investigated as a potential alternative to the quartz crystals. Efforts over recent years have shown that MEMS based oscillators are capable of high Q [9], low temperature drift [10,11], excellent phase noise performance with low power consumption [12], and show almost no long-term drift when packaged [13]. Please note that all these achievements have been published after the start of this work in April 2005. MEMS resonators offer exceptional possibilities compared to quartz crystals regarding miniaturization, form factor, price, and system integration. However, so far, MEMS devices have experienced obstacles and limitations in the quest toward being able to replace crystal devices entirely in timing functions. These issues include power efficiency, excessive system noise and jitter (phase noise), and accuracy. In recent years, the MEMS start-ups SiTime, Silicon Clocks, and Discera have started selling commercial MEMS oscillators to customers. All of these companies have developed their own MEMS oscillator concept to tackle the four P’s. The coming years will show which of these start-ups has developed the most successful concept. In the mean time, also traditional semiconductor companies like ST or NXP are working on MEMS development. In a market report by Databeans [8] a large growth in the MEMS oscillator market is expected over the next years, as is depicted in Fig. 1.5. The forecasted ASP of MEMS resonators is high, due to the anticipated small size and form factor of the timing device.

1.3 PROJECT

DESCRIPTION

1.3.1 NXP Semiconductors

This PhD on design has been performed at NXP Semiconductors. NXP, established in 2006, is a leading semiconductor company. It was founded by Philips, based on their semiconductor division that was created more than 50 years ago. Based in Eindhoven, the company has about 30000 employees working in more than 30 countries and posted sales of $5.4 billion in 2008 [14]. The company is divided into four business units: Automotive & Identification, Home, Multimarket Semiconductors and NXP Software.

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Semiconductors Nijmegen has been founded in 1953. It is the oldest and also the largest manufacturing site of NXP Semiconductors. The most modern facility is the 8-inch wafer foundry. With 12.000 square metres of clean room (equivalent to almost 50 tennis courts), the foundries capacity is of around 450.000 wafers/year [15].

8-inch fab

FIGURE 1.6 – A birds view of the NXP Semiconductors manufacturing site in Nijmegen. The new 52 degrees building had not been build at the time this picture was taken.

1.3.2 Project organization

At NXP Research, the MEMSXO project aims to develop MEMS oscillators on 1.5 μm SOI substrates. The strategic choice for thin SOI substrates has been made for two reasons. First, MEMS processing in thin silicon layers can be done with standard CMOS processing tools. The silicon dioxide layer serves as a sacrificial layer. Second, identical substrates are used for the Advanced Bipolar CMOS DMOS (ABCD) IC-processes. This class of processes can handle high voltages (ABCD2 up to 120V). The high voltage capability is suitable for the transduction of the mechanical resonator. Both MEMS and IC are processed on an identical substrate, since the aim is to integrate the MEMS structure with the IC-process.

The development of the MEMS oscillator is divided into three subjects: MEMS resonator design

Circuit design Processing

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Processing of the MEMS resonators is done at NXP Research in Leuven and Eindhoven. This not only involves the processing of the resonator device, but significant effort is also devoted to the development of a thin-film capping layer. The IC Lab in Eindhoven performs circuit design. MEMS resonator design is both done at NXP Research Eindhoven as well as at NXP Nijmegen. This is the main topic of this thesis.

This PhD on design is performed as an integral part of the MEMSXO project. The bulk of the work is carried out at NXP Nijmegen, although significant time has been spent at NXP Research in Eindhoven to collaborate with the project members there. Other important stakeholders are the Eindhoven University of Technology and the PointOne project MEMSLand. MEMSXO is one of the business carriers in MEMSLand, a MEMS initiative financed by the Dutch Ministry of Economic Affairs. Moreover, this ministry supports this work as a C12 cluster project.

1.3.3 Goal and deliverables

The goal of the MEMSXO project is to show the feasibility of high-performance MEMS oscillators in 1.5 μm SOI. The project addresses specific business cases from business lines in the NXP organization. The requirements of the oscillator will depend on the application in the business case. Most demanding applications are found in wireless communication like GSM or Bluetooth. If the project is able to demonstrate feasibility, the business line can decide to start product development for the specific application. This would mean that the MEMSXO is successful and a technology transfer takes place. The project will then move into a next phase.

This PhD thesis addresses the feasibility of scaling MEMS resonators/oscillators to higher frequencies. The main target frequency for the MEMSXO project is the GSM frequency of 13 MHz. For these platforms, a trend to higher frequencies is observed. Therefore, a 52 MHz platform is anticipated in the future. In this thesis we address the scaling of MEMS resonators to this frequency. We identify the issues concerned with frequency scaling and propose solutions if possible. The deliverables of this thesis on design are:

1 A capacitive 52 MHz MEMS resonator on 1.5μm thin SOI. 2 A piezoresistive 52 MHz MEMS resonator on 1.5μm thin SOI.

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1.4 CONTENTS

OF

THESIS

This thesis consists of eight chapters. We have come to the end of Chapter 1, where we have given an introduction into MEMS and defined the goal and deliverables of this thesis. Chapter 2 provides a technical introduction into MEMS resonators. The processing of the MEMS resonators on SOI wafers is discussed first. Next, the transduction principle of both the capacitive and piezoresistive MEMS resonator is explained. The chapter ends with an overview of the MEMS resonator designs, the electrical equivalent models, and the measurement set-up. This chapter provides the framework for the rest of the thesis.

In Chapters 3 to 6 the main results of the thesis are presented. In Chapter 3, frequency scaling of capacitive and piezoresistive MEMS resonators is discussed. MEMS resonators have been designed from 10 to 400 MHz. The results for the two transduction principles are compared. Chapter 4 shows the results obtained with resonator designs compensated for geometric offset. During processing, inaccurate pattern transfer of the resonator layout causes geometric offset in the real resonator. This results in spread of resonance frequency. The frequency of compensated layouts is in first order unaffected by geometric offset and highly accurate resonators can be obtained. This chapter discusses the compensation method and the results from measurements and FEM simulations. Chapter 5 is devoted to temperature drift of MEMS resonators. Since the stiffness of silicon is temperature dependent, the resonance frequency of silicon MEMS resonators notoriously drifts with –30 ppm/K. This chapter presents results obtained with two compensation techniques: oven-control and thermal oxidation. In chapter 6, a 56 MHz piezoresistive MEMS oscillator is demonstrated. This two-chip demonstrator consists of a dogbone MEMS resonator and an IC amplifier designed in the ABCD2 process.

The thesis concludes with an evaluation of the demonstrated project results in Chapter 7. Next to the evaluation, we put the results obtained in perspective. The NXP oscillator concept that has been developed is compared to other oscillator concepts like LC oscillators, quartz oscillators, and other MEMS oscillators. Finally, concluding remarks are given in Chapter 8.

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CHAPTER 2 FUNDAMENTALS OF MEMS RESONATORS

2.1 INTRODUCTION

This chapter gives a technical introduction into MEMS resonators. The field of MEMS covers the areas of mechanics, electronics, chemistry, and transport phenomena. This chapter will serve as a framework for the following, more detailed, chapters that will frequently refer back to this chapter.

We first discuss the MEMS resonator process. Next, the mechanical model of the MEMS resonator is derived. Specific attention is paid to the Q-factor. Mechanical resonators are commonly preferred over electrical resonators due to their high-Q. After this section, we step into the electrical domain. We start with the transduction principles followed by the electrical equivalent models of the devices. The electrical model for the both the piezoresistive as well as the capacitive resonator will be discussed. Next, the resonator designs utilized in this thesis are presented. The final section covers the measurement set-up. MEMS resonators are characterized with S-parameters, commonly used in high-frequency measurements. For an extended introduction into MEMS topics, the book by Senturia [16] is suggested.

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2.2 MEMS

PROCESSING

The MEMS resonators are processed on silicon-on-insulator (SOI) wafers. SOI is convenient to use, since the silicon dioxide layer can be used as a sacrificial layer. SOI wafers consist of three layers, namely.

The top-layer is the active single-crystal silicon (Si) layer in which devices are made, the SOI layer.

The second layer is an insulating layer of silicon dioxide (SiO2), the buried oxide

layer.

These two layers are on top of a thick single-crystal silicon layer, the handle wafer.

The MEMS resonators at NXP are processed on 8-inch wafers, with a SOI layer thickness of 1.5 μm and a buried oxide layer thickness of 1 μm. The thin SOI layer can be processed with standard CMOS processing tools. A thicker SOI layer would require specialised equipment like a deep-reactive-ion-etcher (DRIE) that significantly enhance process complexity. MEMS resonators in thin SOI allow for batch production in an IC factory.

The process steps are described in Fig. 2.1. A difference is made between open resonators and capped resonators. Open resonators are not enclosed or covered, and are therefore highly susceptible to particles in the surrounding atmosphere. Due to the small volume of the resonators (102-105 μm3), oxidation of the surface or sticking of particles to the surface can significantly degrade the resonator performance [17]. Moreover, as we will discuss in the next section, the resonators operate in vacuum. This should make the need for a cap clear. The MEMS resonators need a clean, low-pressure environment. A cap that encloses the resonator in a cavity provides this. There are a number of methods to cap a MEMS structure:

Wafer-to-wafer bonding [18,19]. Die-to-wafer bonding [20]. Thin-film package [21].

Of these three methods, the thin-film package is the preferred, but also the most difficult solution. The costs of a thin-film package are low, it has minimal height, and the resonators

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are capped early in the process. A successful thin-film package is demonstrated by SiTime [13], where the resonator is covered by a polysilicon cap. At NXP, work is done on a silicon nitride cap. Despite recent success, all results presented in this thesis have been obtained with open MEMS resonators.

To return to the process, the first step is to prepare the lithography masks. The mask drawings are made by the device designer. With the mask pattern, the designer determines the trench pattern that defines the mechanical structure, the aluminium leads and bondpads, the doped and undoped regions of the device, and the areas that need to be covered by the cap. The pattern on mask can be transferred to a resist layer with the help of lithography tools. Selective processing is done on areas that are not covered by resist.

Trenches and holes are etched in the silicon with reactive-ion-etching (RIE). The minimum trench width that can be patterned with the available tools at NXP is 200 nm. Later in this chapter we will see that this minimum trench width is crucial in the transduction principle of the resonator. The maximum trench width is determined by the sacrificial aluminium that defines the cavity. If the trench width is larger than 800 nm, the aluminium fills the trench instead of covering it. Trenches and holes thus have a dimension in the range 200 -800 nm.

The silicon structures are released with a HF vapour etch. The etchant has a high selectivity for silicon dioxide. A vapour etch is used to avoid stiction. This issue is circumvented by using HF vapour, which is capable of high rates. For more detail on MEMS resonator processing, reference is made to internal documentation [22].

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FIGURE 2.1 – The resonators are processed on 1.5 μm SOI wafers. Processing of open MEMS is described in four steps: (a) well implantation (dotted), (b) patterned trench etch to define the mechanical structure and well dope diffusion, (c) contact implantation (hatched) and metal deposition, and (d) sacrificial SiO2 layer etch. The processing of capped resonators deviates from the

open MEMS route after the trench etch. The silicon nitride capping route is described in the following steps: (e) sacrificial SiO2 layer etch to release the MEMS and sacrificial aluminium deposition for cap,

(f) nitride cap deposition (2 times) and plug hole etch, (g) sacrificial aluminium etch, and (h) plug metal deposition and final nitride deposition. This creates a thin-film capped MEMS resonator structure. The deposition pressure of the plug metal determines the lowest achievable pressure inside the cavity. Open Capped

a

b

e

g

f

h

c

d

1.5 μm silicon 1.0 μm silicon dioxide Silicon substrate resist

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2.3 MECHANICAL

RESONATOR

2.3.1 Lumped-element model

A mechanical resonator can be modelled as a simple mass-spring system (Fig. 2.2). To account for energy loss in the resonator a damper is added to the system. The equation of motion for this system is given in 2.1, where x is the displacement (positive or negative) from the equilibrium position, k is the spring constant, m the lumped mass, and γ the damping coefficient. kx x x m F_ext = + +

∑

&& γ& (2.1)

We solve this second-order differential equation for the case where the system is subject to a harmonic force. For this, we assume a harmonic solution for the displacement x of the same angular frequency ω as the external force, but not necessarily of the same phase. We therefore introduce a phase difference φ between the actuation force and the displacement [23]. In the solution for the equation of motion we use two system parameters: the eigenfrequency ω0 and the quality factor Q.

γ ω km Q m k = = 0 (2.2)

The eigenfrequency is a natural frequency of the system for which the vibration amplitude shows a large increase. The Q-factor is a measure of the frequency selectivity of the system. We now solve the equation of motion [23] for the harmonic amplitude x0 and the phase φ.

(

)

(

)

⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − = + − = 0 0 2 2 0 2 2 2 2 0 0 0 1 arctan ω ω ω ω ϑ ω ω ω ω Q Q m F x (2.3)

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From Eq. 2.3 we can see that the amplitude has a maximum for ω= ω0. Actually, the value

is slightly lower than ω0, because of the damping. For high Q-factor and thus low damping this difference is negligible.

The amplitude is 90° out of phase with the driving force at resonance. This means that the driving force is exactly in phase with the velocity and thus the damping force. At resonance the spring force and the inertial force cancel each other and the driving force is used to overcome the damping forces on the system. The vibration amplitude increases with a factor Q compared to the static amplitude

m

k

_γ

FIGURE 2.2 – A simple model of a mechanical resonator consists of three elements: a mass (m), a spring (k) and a damper (γ). Such a mechanical resonator will give a frequency response as depicted in figure E.1. The resonance frequency and the Q-factor of the system can be calculated from the three elements.

2.3.2 Q-factor

The Q-factor has already been introduced in the previous paragraph as a measure of selectivity of the system. It can be calculated from the three mechanical lumped-elements. It can also be derived from the measured resonance peak, where the Q-factor is defined as the ratio of the resonance frequency and the peak width at half maximum. Thus, the sharper the peak, the higher the Q-factor.

(

dB

)

f f Q 3 0 − Δ = (2.4)

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The resonator can also be seen as an energy storage device. In a classical resonator, energy is constantly transformed from one form into another. In a mechanical resonator (i.e. MEMS resonator) energy is transformed from kinetic energy (movement) into potential energy (spring energy). During transformation energy is lost. The Q-factor relates to the amount of energy loss (damping) in the system. Next to the first definition of selectivity (Eq. 2.4) it can also be defined as:

cycle per dissipated energy device in stored energy Q= (2.5)

For every contributing loss mechanism a factor can be determined. The resulting Q-factor of the whole system is the reciprocal sum of all these different Q-Q-factors. This means that the smallest Q-factor is the dominant factor for the overall Q. There are various damping mechanisms that induce energy losses in the system.

Air damping losses (other surface losses) Anchor losses

Thermo-elastic losses

Other intrinsic losses (e.g. non-linearity)

2.3.3 FEM simulations

Next to analytical expression, finite element (FEM) simulations are commonly used to extract the resonance frequency of the MEMS device. The eigenfrequency mode is selected that solves the translational equation of motion. If we assume a harmonic solution u(r,t)=Re(u(r)ejwt), then this equation becomes [24]:

(

_∇

)

₋ 2 ₌0

⋅

∇ c u_i ρω_iu_i (2.6)

Here c describes the stiffness tensor, ρ the mass density, and u(r,t) the particle displacement field.The structure will have an infinite number of eigenmodes, so the subscript

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i points to the ith eigenmode. This equation is solved and from the simulation the eigenmodes and corresponding eigenfrequencies are obtained. Before the simulation can be done, the user needs to set the material properties and boundary conditions.

For the material properties, the density and stiffness of silicon are required. The density is set to 2300 kg/m3_{, a value found in numerous articles on silicon resonators. For the stiffness,}

the two dimensional stiffness tensor is used, since multi-crystaline silicon is an anisotropic material [25].

[ ]

⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⋅ = 80 . 0 0 0 0 0 66 . 1 64 . 0 64 . 0 0 64 . 0 66 . 1 64 . 0 0 64 . 0 64 . 0 66 . 1 1011_Pa C (2.7)

Next to the eigenfrequency, the effective stiffness and mass can also be extracted from the eigenfrequency simulation. For the undamped resonator we know that potential energy is periodically converted into kinetic energy and vice versa. At time t=0, the kinetic energy is zero and all energy is stored as elastic energy.

∫

= = = V s i i el tot E k x W dV E _,_max 2 2 1 (2.8)

with Ws the strain energy densitym and V the volume of the device. From this energy relationship we can extract the effective stiffness and the effective mass. First, we need to determine the vibration amplitude xi. We use a boundary integration in Comsol over the total actuation line. Next we integrate the strain energy over the whole volume to get the total stored strain energy. With the total stored energy and the effective amplitude we can determine the effective stiffness. The effective mass then follows from the resonance frequency and the effective stiffness [26].

2 2 2 4 2 i i i V s i i f k m dV W x k π = =

_∫

(2.9)

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2.4 TRANSDUCTION

PRINCIPLE

2.4.1 Electrostatic actuation

In this project, we use an electrostatic force Fel to actuate the resonator. Electrostatic actuation in combination with capacitive detection is commonly utilized in MEMS resonators, since it can be easily implemented on chip. Piezoelectric actuation is also found often. More exotic materials are needed, however, that increase manufacturing complexity.

To explain how the electrostatic force drives the resonator, an example of a single-clamped beam is given in Fig. 2.3. A superposition of a bias voltage Vdc and a signal input voltage vac are applied on the electrode on the right side of gap g. This induces an electrostatic force on the resonator.

To calculate the electrostatic force, we use a parallel–plate approximation. The parallel plate is formed by the outer end of the resonator beam and the outer end of the electrode that are separated by gap g. For the voltage-controlled moveable-plate capacitor, we take the derivative of the potential energy to calculate the electrostatic force [16,23]. The displacement x of the resonator is defined such that a positive displacement decreases the gap width.

(

)

(

)

2 2 0 2 , x g AV g g V E F V − = ∂ ∂ − = ε (2.10)

Now, the electromechanical coupling coefficient η is introduced. The parameter η describes the coupling between the electrical and the mechanical domain. With η we arrive at the final expression for the electrostatic force. For this, we assume that the displacement x is much smaller than the electrode gap width g.

(

)

2 0 2 0 g V A x g V A V x C v F dc el dc el dc ac ε ε η η ≈ − = ∂ ∂ ≡ = (2.11)

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The coupling factor η is a key parameter. It determines how effective the ac driving

voltage vac (input signal) is coupled as a force on the mechanical resonator. The coupling can be increased by:

Increasing the electrode area Ael. In the example in Fig. 2.3 this means increasing the thickness of SOI layer or increasing the width of the electrode and the beam.

Increasing the bias voltage Vdc.

Decreasing the actuation gap g between the resonator and the electrode.

2.4.2 Frequency tuning with bias voltage Vdc

The electrostatic force is non-linear, since it depends on the displacement x of the

resonator (Eq. 2.11). In the previous paragraph we eliminated the non-linear terms by assuming that x << g. If we use a Taylor expansion for the term 1/(g-x)2 instead and neglect all higher terms of x/g with the same assumption, we find an actuation force that is linearly dependent on displacement [23]. If we use this expression for the force in the equation of motion (2.1), we find an equation for the frequency tuning with the dc bias voltage.

3 0 2 0 2kg A V f f _≈ − _dcε _el Δ (2.12)

Bias tuning of the frequency can be used to trim the frequency after processing. This is a large benefit and can be exploited by designing resonators that have a large tuning range. This can also be a disadvantage, however. If the resonance frequency is highly sensitive to the bias voltage, this will set high demands on bias voltage stability of the product. This is a clear trade-off and the optimal choice will depend on specifications. Bias tuning range can be increased by:

Increasing the electrode area. Decreasing the gap width.

Decreasing the effective stiffness.

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2.4.3 Capacitive detection

For capacitive resonators, a capacitive current is used to detect the mechanical resonance. Figure 2.4 shows a top view of the 3D resonator in Fig. 2.3. When the bias voltage is applied over the parallel plate capacitor, a charge Q of opposite sign is induced on the two plates. The amount of charge that is induced on the plates depends on the bias voltage, the electrode area and the gap between the two plates. The displacement x has again been defined such that a positive displacement decreases the gap width.

If the two plates are moving towards each other, more charge is induced on the plates and current flows towards the outer end of the beam. If the plates are moving away from each other, less charge is induced on the plates and current flows away from the outer end of the beam. This ‘motional’ current is thus induced by vibration motion of the mechanical resonator that causes a capacitance change between the two plates.

The detected current is a superposition of the motional current and the electrical current induced by the ac voltage over the static gap capacitance. The motional current is proportional to the velocity of the mechanical resonator, where we neglected the ac voltage assuming that Vdc >> vac. Again, η is the key parameter in the coupling between the

mechanical resonator and the electrical output signal.

t v C t x t V C t x x C V dt dQ I ac dc _∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ ⋅ ∂ ∂ = = η (2.13) 2.4.4 Piezoresistive detection

For the piezoresistive resonator, actuation is again done electrostatically. To detect the mechanical resonance, a modulated resistance is used instead of a modulated capacitance, as discussed in the previous paragraph. To describe this readout principle, we stick to our example of the single-clamped beam. We have a look at the top view of this device, which is presented more schematic this time (Fig. 2.5).

The vibration of the resonator induces strain εL=ΔL/L in the material. This strain induces

a proportional change in resistance, as is described by Eq. 2.14. This proportionality is defined in the gauge factor K. The gauge factor describes both geometrical induced changes and resisitivity (ρ) induced changes.

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Si SiO2 Si substrate + -+ -Vdc vac

~

Fel g

FIGURE 2.3 – The 3D structure of the single-clamped beam resonator. The SOI wafer consists of three layers: the active silicon (Si) layer, an insulating silicon dioxide layer (SiO2), and a thick bulk

silicon layer that is only partly shown in the figure. The resonator is actuated with the alternating electrostatic force ΔFel over gap g due to input voltage vac.

-Q + + + + +Q g iac x + -+ Vdc vac

~

-FIGURE 2.4 – Capacitive readout of the single-clamped resonator. The figure shows a top view of Fig. 2.3. The bias voltage induces a charge Q of opposite sign on both sites of the gap. If the resonator vibrates the gap size is changed by the displacement x. This changes the amount of charge on both plates and induces a current in the resonator.

Vdc + vac

g x

L Id

FIGURE 2.5 – Piezoresistive readout of the single-clamped resonator. Again, a top view of the resonator is shown. A slit in the beam forces the current Id along the length L of the beam. The

vibrational motion of the resonator induces strain in the material. This changes the resistance of the current path over the resonator. This modulated resistance is read out with the dc current that induces an ac voltage component over the resonator.

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(

)

A L A L R K R R L ρ ρ ε = = Δ , , (2.14)

For non-piezoresistive materials, the resistivity does not change under strain. The change in resistance is due only to the geometrical effect. For a Poisson ratio of 0.5 this results in a gauge factor of 2. Actual Poisson ratios will be even smaller than 0.5.

For piezoresistive materials, like silicon, gauge factors are two orders of magnitude larger than this. The gauge factor is dominated by the change in resistivity and the geometrical effect can be neglected [27]. This makes piezoresistive readout attractive to use in silicon.

It is straightforward to calculate resistance change for the single-clamped beam in Fig. 2.5. The only assumption that is made is that the slit spans the entire length of the device. The solution shows that the amount of resistance change is proportional to the mechanical vibration amplitude x. Since the resonator describes a harmonic motion, a harmonic resistance modulation is induced.

x L K dx L K R R L = = Δ = Δ

∫

0 ε ρ ρ (2.15)

To detect the resistance modulation, a dc current Id is applied over the resonator. This leads to a modulated voltage vd superimposed on a dc voltage Vd. The capacitive resonator is a passive device, but the piezoresistive resonator is an active device, since dc power is consumed in the resonator. Higher output voltage can be induced at the expense of higher power consumption.

2.4.5 Frequency tuning with bias current Id

With the piezoresistive readout there is a second method of frequency tuning. Young’s

modulus E of silicon, and thus the stiffness of the structure, drifts with temperature at about -60 ppm/°C. If a current is applied through the resonator, dc power is dissipated in the

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lower values. The current Id can thus be used to tune the resonance frequency. The dominant heat loss mechanism is conduction via the suspension of the structure. The thermal isolation of the suspension therefore determines how effective the frequency can be shifted with Id. This oven-control principle will be discussed in more detail in chapter 5.

Lm Cm

Rm

Cs

FIGURE 2.6 – The parallel capacitance Cs is added to the electrical equivalent model of the MEMS

resonator. Unlike the motional parameters Cs is a real capacitor formed by the gap spacing between

the electrodes and the resonator.

52,20 52,24 52,28 -110 -100 -90 f [MHz] ma gY [d B] 0 90 pha seY [ ° ]

FIGURE 2.7 – The figure shows the measured admittance magnitude (∆) and phase (□) of a 52 MHz capacitive MEMS resonator. The admittance has been extracted from calibrated S-parameter measurements. The data is fitted (–) with the electrical equivalent model in Fig. 2.6. A frequency selectivity of 7 dB has been found for this device.

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2.5 ELECTRICAL EQUIVALENT MODELS

2.5.1 Capacitive resonators

In the previous section the transduction of the MEMS resonators is discussed. For the capacitive resonator we learned that the force on the resonator is proportional to the input voltage (Eq. 2.11) and the output current is proportional to the velocity of the resonator (Eq. 2.13). In both equations the electromechanical coupling coefficient η describes how effective the electrical and mechanical domain are coupled. Now, we will use the two equations to rewrite the mechanical equation of motion of the mass-spring-damper system (Eq. 2.1).

q k q q m v_ac ₂ ₂ ₂ η η γ η + + = _&& _& (2.16)

This differential equation is identical to the differential equation of an RLC-network. We can therefore model the electrical response of the resonator by an RLC-network. Each mechanical lumped element has a direct equivalent electrical lumped element. The R, L, and

C all have a subscript m, since they are called the motional parameters.

⎪ ⎪ ⎪ ⎪ ⎩ ⎪⎪ ⎪ ⎪ ⎨ ⎧ = = = = k C m L Q km R m m m 2 2 2 2 η η η η γ (2.17)

The RLC-network (Fig. 2.6) models the capacitive current that is induced by the mechanical movement of the resonator. These motional parameters are not real electrical parameters, but electrical equivalent parameters. Off-resonance the MEMS device behaves as a capacitor that is made up by the gap spacing between the resonator and the electrodes. This is a real electrical parameter and it is a signal path, parallel to the RLC-network, from input to output. To calculate the value of Cs, we use the parallel plate approximation.

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g A C el s 0 ε = (2.18)

At series resonance, the effect of Lm and Cm cancel out. The impedance is then determined by the parallel combination of Rm and Cs. This means that with the addition of Cs, the phase at series resonance is no longer zero. Furthermore, the parallel combination of Cs and Lm (and

Cm) induces a parallel resonance. At series resonance the impedance is minimum, while at parallel resonance the impedance is maximum. For our MEMS resonators, Cs is three orders of magnitude larger then Cm. This means that the parallel resonance approaches the series resonance.

For a resonator, frequency selectivity is determined by the Q-factor. For our MEMS resonators, however, we have to take the influence of Cs into account. The ratio of Rm and the impedance of Cs determine how high the resonance peak extends over the background signal. We will take this as our definition for frequency selectivity Δs. The frequency selectivity of

the capacitive resonator is indicated in Fig. 2.7.

m s s R C f₀ 2 1

π

= Δ (2.19) 2.5.2 Piezoresistive resonator

For the piezoresistive resonator, we can either apply a voltage Vd over the resonator arms or a current Id through the resonator arms. When we apply a fixed voltage over the resonator, we measure a modulated current. With a fixed current through the resonator, we measure a modulated voltage. The choice is arbitrary and both lead to similar small-signal models [28].

All measurements in this project have been performed with a fixed current through the resonator. In Eq. 2.20, the output voltage vd is derived. This is a function of the detection current modulation id and the voltage gain µm of the input voltage vg.

g m d d g i g d d v d d d d d v i R v V V i I V v R R R I V d g μ + = ⋅ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⋅ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Δ + = 1 (2.20)

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For the piezoresistive resonator, the voltage over the gap Vg is different from the bias voltage applied on the electrodes Vdc. The current through the resonator lifts the potential of the resonator. If we assume that the outer end of the resonator (opposite to the electrode) is halfway along the current path, then Vg=Vdc-IdRd/2.

Equation 2.20 incorporates a voltage gain μm. For the small signal model of the piezoresistive model we choose to use a transconductive element instead, to follow previous publications on this subject. This model has been developed by Van Beek [29]. Following from Thévenin’s theorem and Norton’s theorem, the use of a voltage gain or a transconductance is equivalent. We only have to take the 180° phase shift into account [28]. The electrical behaviour of the resonator is identical to a field-effect transistor in the linear region, with frequency selectivity. With the use of the field-effect transistor model, the subscript g at the input side now stands for both the electrode gap and the gate of the transistor. Accordingly, subscript d at the output side can be read as detection or drain. The transconductance gm is given by:

⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⋅ ⋅ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + − = tot geom el d g m m m R R g k L A E K I V g Q j g g mod 2 2 0 0 0 2 0 2 0 4 1 π ε ω ω ω ω (2.21)

For the piezoresistive resonator, the capacitive signal, explained in paragraph 2.5.1, is superimposed on the piezoresistive readout. In most cases, at resonance, the piezoresistive signal is a few orders of magnitude larger than the capacitive signal. For low currents Id or high gap voltages Vg, the magnitudes of both signals are comparable. Moreover, the capacitor

Cs dominates the response of the piezoresistive resonator off-resonance. Therefore, the capacitive branch (Fig. 2.6) is added to the small signal model of the piezoresistive resonator. The small signal model is depicted in Fig. 2.8. Frequency selectivity Δs of the piezoresistive

resonator is indicated in Fig. 2.9 and can be calculated with Eq. 2.22.

s m s C f Q g 0 0 2

π

= Δ (2.22)

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Rd/2 gmvg gmvg Cs Rm Lm Cm vd vg Rd/2

FIGURE 2.8 – The complete small signal model for the piezoresistive resonator with the capacitive branch incorporated into the model. This model is for a piezoresistive resonator where a current is applied through the resonator arms.

52,98 53,00 53,02 -120 -100 -80 f [MHz] ma gY [d B ] 0 90 180 ph ase Y [ ° ]

FIGURE 2.9 – The figure shows the measured admittance magnitude (∆) and phase (□) of a 52 MHz piezoresistive MEMS resonator. The admittance has been extracted from calibrated S-parameter measurements. The data is fitted (–) with the electrical equivalent model in Fig. 2.8. A frequency selectivity of 23 dB has been found for this device.

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2.6 RESONATOR

DESIGN

For the design of the MEMS resonators, a choice for bulk-acoustic-mode resonators over flex-mode resonators has been made. Bulk-acoustic resonators have a higher effective stiffness compared to flex-mode resonators. This leads to some major advantages, especially at higher frequencies like 52 MHz, which is the main target frequency for this project.

High energy storage. This is important to achieve a high signal-to-noise ratio. Higher Q-factor.

Larger electrode area (size in general).

Although an in-plane flex-mode design is not attractive at 52 MHz, good results have been obtained with an out-of-plane design [30]. This would mean a change in processing, however, which lies beyond the scope of this project.

Round shapes, like rings or discs often found in literature, are also excluded from this project. The argument is that device layouts are made on a grid structure. This makes a curved line an ill-defined shape. It will not follow common design rules in the wafer fab. Moreover, a square shape will offer the same benefits as a disc, and can be made according to design rules. ρ λ E v c f s sound res = = (2.23)

The resonance frequency of a bulk-acoustic resonator can be calculated with Eq. 2.23, where csound is the velocity of the longitudinal sound wave propagating in the material. The

wavelength depends on the size of the resonator and the mode number we choose to actuate. In all measurements, the resonator is actuated in its fundamental mode.

For the square plate resonator, the outer ends in both directions are free. This means that in the fundamental breathing mode (Fig. 2.11a), a standing wave of half a wavelength propagates in the material. This is true for both in-plane dimensions, so the displacement pattern of the square plate is the superposition of the two orthogonal longitudinal waves.

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L s w Free-free beam b a L s w square hole size h Dogbone L square hole size h Nh holes is each

row and column

Square plate

FIGURE 2.10 – Schematic top view of the free-free beam, dogbone, and square plate resonator. Geometrical parameters are indicated in the figure.

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For the resonance frequency of the plate, not taking holes and corner suspension into account, we find (Fig. 2.10): ρD E L f 2 0 2 1 = (2.24)

Silicon is a anisotropic material, so Young’s modulus depends on the crystallographic orientation. For the square plate, we use an effective 2D Young’s modulus of 181 GPa [25]. Because of its large stiffness and volume, the square plate is able to store a lot of energy. This means that this design is well suited to meet demanding phase noise specifications [31,32]. The square plate in breathing mode is actuated by electrodes on all four sides. With this large electrode area the motional resistance Rm can be kept low, despite the large stiffness of the structure.

The square plate is also frequently actuated in Lamé-mode (Fig. 2.11b), which is a shear mode. The Lamé-mode is attractive, since the four corners are nodal points of this mode. The structure can thus be suspended at its nodal points, allowing very high Q-factors. The resonance frequency of this mode is not determined by the Young’s modulus but by the shear modulus G, which is 160 GPa for silicon.

ρ G L f 2 1 0 = (2.25)

For the free-free beam resonator both outer ends are also free. The fundamental mode therefore also consists of half a wavelength (Fig. 2.11c). However, the free-free beam is defined as two parallel resonators connected back-to-back for a balanced operating mode [33]. The bridge in the middle suspends the structure at the nodal point (Fig. 2.10). The resonator thus consists of two beams of length L that both contain a quarter of a wavelength in fundamental mode. Because of the balanced operating mode, the free-free beam design is capable of very high Q-factors. Since the design is small, it is easy to process and cap. The design is oriented along the [100] direction that has a Young’s modulus of 131 GPa.

ρ ] 100 [ 0 4 1 E L f = (2.26)

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FIGURE 2.11 – FEM simulations show the eigenmode shape of the square plate (2), free-free beam, and dogbone resonators. The colours correspond to total in-plane displacement, where dark red is the maximum displacement and dark blue no displacement. Typical displacements for the MEMS resonators are a few nanometre. In figure (a) and (b) the square plate design is depicted. For the same square size, the extensional or breathing mode is depicted in (a) and the Lamé mode in (b). The plate is perforated with regular spaced square holes to enable the etch of the SiO2 layer underneath the Si

structure. In figures (c) the free-free beam and in (d) the dogbone resonator are depicted. Both designs have a slit to force the current along the length of the structure. This is used for the piezoresistive readout, since for the selected mode shapes most strain is along the length.

a: 52 MHz, breathing mode square plate 80 µm

b: 41 MHz, Lamé-mode square plate

c: 52 MHz, piezoresistive free-free beam d: 52 MHz, piezoresistive dogbone 26 µm

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The most complex structure is the dogbone resonator (Fig. 2.11d). It is very similar to the free-free beam, but with a lumped mass attached to the outer end [34]. This lumped mass has two benefits; it increases the electrode area and it concentrates the strain in the springs. Because the dogbone has a non-uniform cross section, the fundamental mode consists of two standing waves that connect at the step in cross-section. The resonance frequency can be found with equation 2.25, which has been derived by Van der Avoort [35]. The parameters w and L correspond to the width and length of the spring and b and a to the width and length of the mass.

( ) ( )

] 100 [ 2 0 2 2 4 tan tan 2 E f a L b w ρ π β β β = = (2.27)

2.7 MEASUREMENTS

2.7.1 Two-port S-parameter measurements

MEMS resonators, which are the devices-under-test (DUT) in the measurements, are two-port devices. The MEMS resonators are measured with a network analyser, which measures scattering parameters. The name scattering comes from the similarity between the behaviour of light in the field of optics and microwave energy. S-parameters [36,37] provide a way to give a complete description of the electrical behaviour of the DUT. Since electrostatic MEMS resonators are reciprocal devices, S12 and S21 are equal, thus 3 independent complex

numbers can be measured. The main reason to use S-parameters is that they are measured in a matched impedance system (in this case 50 Ω), in contrast to open- and short-circuit-type measurement required for other available network parameters. Measurements that are rather difficult to make at microwave frequencies. Furthermore, when measuring S-parameters with a network analyser, effects of cables, bias tees and the network analyser itself can largely be eliminated.

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2.7.2 Measurement setup

A custom-designed setup is used for device characterisation. The MEMS resonators are measured in vacuum, since the resonator movement is severely damped at ambient pressure. For vacuum measurements, a custom-made stainless steel vacuum chamber is used (Fig. 2.12a). Air pressure inside the chamber is reduced by a Pfeiffer turbo molecular drag pumping station with a membrane pre-pump that has a base pressure of 5·10-5 mbar. For resonator measurements, a pressure of 10-2 mbar is low enough. At this pressure, air damping is no longer the dominant dissipation mechanism. This pressure is reached in about 5-10 minutes. Electrical feedthroughs on the side of the vacuum chamber can be used to connect signal sources outside the chamber to the probes inside the camber.

A CCD camera on top of a microscope is used to probe the bondpads of the device. Input and output of the devices have a ground-signal-ground bondpad configuration with a 100 μm pitch. This fits the Süss mircoTec Z-probes used.

The frequency response of the resonator is measured with an Agilent E5062A ENA-L network analyser. Bias voltage and bias current are applied externally, usually with Keithley 2400 SourceMeters. Bias is applied externally, since the network analyser is only capable of bias voltages up to 30V. Bias tees shield the network analyser from the high dc voltages and currents. The PicoSecond 5530A bias tee used is suited for dc voltages up to 200 V. In addition, a 1MΩ resistor is added to prevent high currents in event of a short.

Both isolated and grounded chucks are available for probing. For all measurements in this project an isolated chuck has been used, since a grounded chuck complicates interpretation of the measurements. The grounded chuck becomes part of your DUT and its influence has to be added to the electrical equivalent circuit, which is an unnecessary complication.

Calibration is needed to exclude effects of cables, probes, bias tees and network analyser from the DUT measurements. For this calibration a specific calibration substrate, Picoprobe #CS-5, is used. A full two-port calibration consists of 12 measurements. After calibration the DUT is measured in vacuum. Next, the model parameters are extracted in ICCAP. The measured amplitude and phase of all four S-parameters are fitted with the electrical models in Fig 2.6 and Fig. 2.8.

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FIGURE 2.12 – The MEMS resonators are measured in vacuum. In figure (a) the vacuum chamber is depicted with the lens and CCD camera used for probing. In figure (b) a Z-probe can be seen that is used to perform the measurements on wafer.

Bias tee Network Analyser 1 MΩ 1 2 + -Bias tee Vdc Id

FIGURE 2.13 – The measurements set-up for measurements on a piezoresistive resonator.

2.7.3 Semi-automatic probe station

For waferspread measurements a semi-automatic probe-station is used. In this case, measurements are done at ambient pressure. One 8-inch wafer consists of ≈140 identical fields. The Labview controlled waferstation is able to scan a complete wafer and measure an identical resonator design in each field. Measurements are done uncalibrated with an older network analyser, the HP 8753D. It takes about 90 minutes to complete a single scan.

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The data is analysed in Matlab. From the uncalibrated data the motional resistance, Q-factor, and resonance frequency can be extracted. Matlab automatically generates 2D waferplots showing the spread of the three extracted parameters over the wafer.

To improve the accuracy of the frequency measurements, the wafer is first heated to 140 °C and fixed at this temperature for an hour. This will remove the thin water film on the resonator surface. After this heat treatment the wafer is cooled down to room temperature and measurements can start. During measurements a flow of dry air is maintained over the wafer, to prevent water molecules to stick on the surface. The resolution is < 500 Hz, which is 10 ppm for a frequency of 52 MHz [38].

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CHAPTER 3 FREQUENCY SCALING

3.1 INTRODUCTION

Micromechanical resonators offer capabilities of having multiple frequency standards on a single-chip. This is especially true for lateral resonators, where the resonance frequency is determined by the design on the lithography masks. In this regard, it is an interesting question to see what frequency range is within reach for MEMS resonators.

The frequency range for oscillators ranges from low kHz to several GHz [39]. The low-frequency oscillators are generally used for real-time clock (RTC) applications, for example the 32.768 kHz RTC found in watches. At the mid-end of the spectrum we find the frequency reference applications, like the 26 MHz found in GSM phones. RF oscillators cover the high-frequency range.

Low-cost ceramic resonators work well in embedded systems in which timing is not critical [40]. They are available at frequencies of 200 kHz to almost 1 GHz. For quartz crystals, higher frequencies are usually obtained at lower cost by using a phase-locked loop (PLL). The PLL consumes power, however, and degrades the phase noise performance of the oscillator.

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TABLE 3.1 – The square plate resonator is geometrically scaled six times a factor two. The parameters refer to Fig. 2.10.

13 MHz 26 MHz 52 MHz

416 MHz 208 MHz

104 MHz

FIGURE 3.1 – SEM pictures show the scaling of the square plate resonator. The resonance frequency is inversely proportional to the size of the square plate. The electrodes appear brighter due to charging. device # L [µm] h [µm] Nh Lel [µm] f0 [MHz] 1 320 0.5 35 307 13 2 160 0.5 17 147 26 3 80 0.5 9 69.5 52 4 40 0.5 5 32.2 104 5 20 0.5 2 13.3 208 6 10 0.5 - 5.0 416

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For a MEMS resonator, higher frequencies actually mean a reduction of cost. For a higher frequency, the resonator has to be reduced in size and that means less silicon area is consumed. Silicon area is a cost driver for the chip. Conversely, decreasing frequency means larger resonators. For thin layer structures (e.g. 1.5 µm SOI), this would mean that robustness is a major issue. For low frequency MEMS resonators, thicker layers are preferred.

In this chapter we will investigate the scaling of MEMS resonators to higher frequencies. We will evaluate whether high-frequency applications are suitable for MEMS resonators. Furthermore, the scaling of the piezoresistive and capacitive concepts will be compared.

3.2 CAPACITIVE

RESONATOR

3.2.1 Design of experiment

The square plate design is used to investigate the scaling of the capacitive resonator. The breathing eigenmode of the resonator is depicted in Fig. 2.11a. The square plate is actuated by four electrodes coinciding with the side ends of the plate. To etch the SiO2 layer

underneath the resonator, the plate is perforated with a regular pattern of square etch holes. The experiment consists of six scaling steps. In each scaling step, the size of the plate and T-shaped suspensions is decreased by a factor two. The electrode length is also decreased, but an exact factor two is not always possible due to design rules and structural restrictions.

The SOI layer thickness (1.5 μm), gap width (270 nm), hole size (500 nm), and pitch (9 μm) are kept constant. This means that the number of etch holes Nh2 scales as well. Table 3.1 and Fig. 3.1 give an overview of the scaling experiment with the square plate resonator. The experimental results presented in the next paragraphs have all been obtained with calibrated S-parameter measurements in vacuum (section 2.7).

3.2.2 Scaling with bias voltage Vdc

As has been explained in chapter 2, according to the analytical model, the three motional parameters are expected to scale with bias voltage (Eq. 2.17). Also the resonance frequency, due to the non-linear electrostatic actuation force, should scale with bias voltage (Eq. 2.12).

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To test this analytical model empirically, devices are measured at various settings of the bias voltage. The resonators are characterized with calibrated S-parameter measurements in the vacuum set-up (section 2.7).

For the 13 MHz square plate, the results are plotted in Fig. 3.2. The electrical parameters are extracted in IC-CAP [41] where the electrical model (Fig. 2.6) is manually fitted to the measured data. The expected linear dependency is found for all four parameters, which shows that the analytical model accurately describes the bias voltage dependency. A motional resistance of 940 Ω has been measured at 80V bias voltage. This is the lowest motional resistance measured in this project and the only time a motional resistance below 1 kΩ has been measured. 0 2 4 6 0 100 200 300 V_dc2 [103 V2] C m [aF] 1,748 1,752 1,756 f o ₂ [10 14 Hz 2 ] 10-4 10-3 10-2 100 101 102 V-2_dc[V-2] L m [H] 100 101 102 R m _[k Ω ]

FIGURE 3.2 – The resonance frequency and the three motional parameters all scale with the electrode bias voltage. In figure (a), Cm and fo2 are plotted against the bias voltage squared. In figure (b), Lm and

Rm are plotted against the inverse of the bias voltage squared. An excellent linear fit is obtained in all

four cases.

3.2.3 Mechanical parameter extraction

From the measurements we are able to directly extract the electrical equivalent parameters. With the four slopes obtained from the bias voltage plots, the mechanical parameters can also be extracted. For this, we need some mathematical rework (Eq. 3.1). The only input parameter not measured is the electrode area Ael.

The electrode area can be calculated from the layout of the device and the SOI layer thickness of 1.5 μm. However, using this value for the electrode area does not yield a consistent parameter set compared to the gap width g extracted from SEM pictures (270 nm) and k and m extracted from Comsol simulations. As a pragmatic solution, we introduce the